Lesson 1.15

Multiplying Complex Numbers

Multiplying complex numbers is just like specific polynomial multiplication, with one special twist: transforms into !

Introduction

We use the FOIL method (First, Outer, Inner, Last) just like with binomials. The only difference is that is not a variable—it has a value.

Past Knowledge

Today's Goal

Multiply and simplify the result.

Future Success

This is crucial for "rationalizing the denominator" (division) in the next lesson.

Key Concepts

1. The Golden Rule of

Whenever you see , you must replace it immediately.

Never leave an exponent on in your final answer.

2. The FOIL Method

Multiply binomials as usual:

Notice the last term changed sign because .

Worked Examples

Example 1: Distributing Monomial

Basic

Simplify .

Distribute

Replace

Standard Form:

Example 2: FOIL Two Binomials

Intermediate

Multiply .

FOIL

Simplify

Result:

Example 3: Squaring Binomials

Advanced

Simplify .

Expand

Do not distribute the exponent! Write it twice.

FOIL & Simplify

Result:

Common Pitfalls

Leaving

An answer with is incomplete. It's like leaving uncalculated.

The "Freshman Dream"

. You must FOIL to get the middle term.

Real-Life Applications

Multiplying by real numbers scales a value (makes it bigger/smaller). Multiplying by rotates it. In the complex plane, multiplication by is a 90-degree turn. This math is used to control rotation in video game graphics and robotics!